Continuity mistake: Hogan dumps a bucket of water on Barnes and Davis each, to make it look like they were the ones who hid in the tanker. They get big wet patches on their uniforms from it, but aren't completely wet through and through - which is to be expected, after all it's just a small bucket of water each. After the next cut however, they are completely soaked like they went swimming.
Continuity mistake: When the man and woman from the underground meet Hogan at the Hoffbrau, they're taking off their outer coats. The man takes off a dark brown coat and drops it. Hogan picks it up, hands it to the guy who hangs it up. The man removes his dark brown overcoat revealing a tan coat. The next shot, he"s taking off the dark brown coat again, hangs it up and goes for beer.
Continuity mistake: After Shultz has been discredited, Klink is taking away the "badges of rank". When Klink goes to crush the monocle Shultz was wearing, he has his monocle on. After the shot cuts to a closeup of Shultz's monocle being crushed, and widens out, you see Klink's monocle is suddenly gone. It remains missing until Klink goes to the hat rack. The shot cuts from Klink, to Hogan and Shultz, then back, and the famous monocle is suddenly back, both without Klink stopping to take it off, or put it back on.
Continuity mistake: Carter and Newkirk have just taken the escapees down the tree trunk to the tunnels. When they go to open it, they clear off the snow to open it. When the shot goes to a German soldier that was chasing them, the phony stump is suddenly covered over again. It would still have been visible, having just been exposed.
Continuity mistake: As Schultz marches in with the Kraut platoon, he reports to Klink. As he goes to stand at attention and report, the new tunnel section he's over collapses, and he completely disappears into the hole. The camera goes to a close up on Klink, then back to the hole, and you see Schultz standing in the hole and it's only chest deep.